AbstractThe role of adaptation in biological invasions will depend on the availability of genetic variation for traits under selection in the new environment. Although genetic variation is present for most traits in most populations, selection is expected to act on combinations of traits, not individual traits in isolation. The distribution of genetic variance across trait combinations can be characterized by the empirical spectral distribution of the genetic variance–covariance (G) matrix. Empirical spectral distributions of G from a range of trait types and taxa all exhibit a characteristic shape; some trait combinations have large levels of genetic variance, while others have very little genetic variance. In this study, we review what is known about the empirical spectral distribution of G and show how it predicts the response to selection across phenotypic space. In particular, trait combinations that form a nearly null genetic subspace with little genetic variance respond only inconsistently to selection. We go on to set out a framework for understanding how the empirical spectral distribution of G may differ from the random expectations that have been developed under random matrix theory (RMT). Using a data set containing a large number of gene expression traits, we illustrate how hypotheses concerning the distribution of multivariate genetic variance can be tested using RMT methods. We suggest that the relative alignment between novel selection pressures during invasion and the nearly null genetic subspace is likely to be an important component of the success or failure of invasion, and for the likelihood of rapid adaptation in small populations in general.